The quartic construction was considered by Riemann as only a formal possibility. It is a quite remarkable fact that this construction can be given a well-posed physical interpretation in electromagnetic wave propagation physics. As it was already known to Gauss and Riemann, the geometry of our world can be probed by the light propagation phenomena, i.e., by the electromagnetic waves. Consequently, the geometric structure can be viewed as only a secondary phenomena that is established by the properties of the media. For details of this premetric approach, see [1]} and [2]}.
